[roblev]

For me the main problem with the "if the demon says TV then I listen to the radio" paradox is that it asks the demon to predict its own computation.

Take as an example COnways game of life. It is perfectly predictable in the future - given current state, I can easily predict the state N steps in the future (I just run the simulation on my side computer). And clearly my side prediction will have no impact on how the game-of-life evolves. I can even insert my prediction into the game-of-life by changing the value of my cell (alive=watch TV, dead=listen to radio). Even if a "perverse" in-game character chooses to do the opposite of my prediction, it doesn't alter the fact that I can perfectly predict future of the game universe.

But if I am asked to make that prediction inside the game of life using a program embedded in the game-of-life, then that is a lot harder. It seems very unlikely that I could encode the state of the universe in a way that allows my prediction program to run in any way more efficiently than just letting the game-of-life play out. I'm pretty sure that the Halting problem could be used to demonstrate this.

So for me it makes a big difference whether you constrain the demon to "in universe" computation or "out of universe" computation.

[espadrine]

To me, there are two more issues with that demon paradox.

First, that it is assumed that its answer has a physical representation in reality. The fact that solving a non-trivial maze from the inside is impossible without moving through the maze doesn't mean the maze cannot be solved without moving. It can be solved from above the maze. Similarly, assuming quantum mechanics is accurate in that every particle has a random nudge in its position and velocity, if we had access to the random generator, we could create that demon; otherwise, the solution would be expressed in terms of what that generator produces: a solution which doesn't have a physical representation, but is merely a probability distribution.

Second, there is this assumption that its answer be crystal clear for us to understand. Any amount of work with symbolic computation gives the intuition that sometimes, the answer isn't atomic. In that case, the demon's answer could be "if I say that you listen to the radio, what you will do is watch TV. Otherwise, you will listen to the radio".

[tel]

That was my thought, too. It makes the Demon impredicative which always produces size issues. It could perhaps state that in a world in which it does not render a response you would choose to do X, but a world with feedback cannot be predicted by this demon.

But then there could be a Laplace's Demon+1 which sits above this all and knows precisely that Laplace's Demon+0 will render a response to that question and Demon+1 still can accurately predict the future.